It's time to get out those party invitations! Trouble is, your sister doesn't get on with your boyfriend, your boyfriend doesn't like your best mate, and your best mate's just broken up with your cousin. Who do you invite? Find out why the answer could win you a million dollars! And for a slightly easier question, can you make sure that no two people you've invited have the same number of friends?

Oh, those beautiful snowflakes! They've put us in the mood for fractals, so let's celebrate some favourite shapes:

Pandora's 3D box
An amateur fractal programmer has discovered a new 3D version of the Mandelbrot set. The new creation is based on similar mathematics as the original 2D Mandelbrot set, but its infinite intricacy extends into all three dimensions, revealing fractal worlds of amazing complexity and beauty at every level of magnification.

Unveiling the Mandelbrot set
And if you're wondering what the Mandelbrot set is to start with, here's an introduction from one of the world's experts.

Modelling nature with fractals
Computer games and cinema special effects owe much of their realism to the study of fractals. This article takes you on a journey from the motion of a microscopic particle to the creation of imaginary moonscapes.

How big is the milky way?
A question which has been vexing astronomers for a long time is whether the forces of attraction between stars and galaxies will eventually result in the universe collapsing back into a single point, or whether it will expand forever with the distances between stars and galaxies growing ever larger. Learn how fractal dimension gives us a way of
approaching the question.

Extracting beauty from chaos
Images based on Lyapunov Exponent fractals are very striking. Find out what Lyapunov Exponents are, what the much misunderstood phenomenon of chaos really is, and how you can iterate functions to produce marvellous images of chaos from simple mathematics.

Non-Euclidean geometry and Indra's pearls
If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane. But once you move into the curved world of hyperbolic geometry, possibilities become endless and the most amazing fractal structures ensue.

As the odds for a white christmas in the UK shorten and we rug up for another snowy weekend, here is the Plus weather forecast...

And now, the weather...
Snowstorms, floods and hurricanes remind us yet again that accurate forecasts are necessary not only to protect property, but more importantly to save lives.

Met office in for another roasting?
We all know that weather prediction, though incredibly sophisticated and advanced, is an inexact science. Perhaps if the Met Office published the probabilities underlying its forecasts, it might get a more rational response to disappointing weather.

How maths can make you rich and famous: Part II
The weather is described by the famous, and infamously difficult to solve, Navier-Stokes equations. And one million dollars is waiting to be won by anyone who can solve this, one of the grand mathematical challenges of the 21st century.

The butterfly flap felt across the world
Edward Lorenz, American mathematician and meteorologist, was known as the father of chaos theory. But he is perhaps best known for coining the term butterfly effect.

You've opened all your presents, you've eaten till you're ready to burst, you're sick of the holiday specials on TV. Let's play a game!

Curious dice
In this article we present a set of unusual dice and a two-player game in which you will always have the advantage.

Games people play
Mathematicians love games. Not only can they have fun while looking like they're busy working, but even the simplest games can demand clever tactics and strategies to win.

Practice makes perfect
Although it may be inevitable that computers will become unbeatable in the near future, human Grandmasters are still holding their own against the machines today. How is it, then, that the human brain, with a mere fraction of a computer's number-crunching ability, is still able to put up a good fight?

Let 'em roll
This article was the winner of the schools category of the Plus new writers award in 2006. Dice are invaluable to many games, especially gambling games, but instead of playing with ordinary 1-6 numbered dice here are two interesting alternatives — with a twist!

And don't forget, you can also read all about how game theory can be used for more than just fun. Good luck!